Why did Babbitt's wife object to him using the word 'tux' in Sinclair Lewis's book?

But from my extremely limited recall, the tuxedo has the shiny stripe in the side of the trousers?

Right, and the tux jacket has satin lapels to match. Wearing either without the other would look very strange.

The ranges of dress gets quite ridiculous.

Black tie aka tuxedo is just one. (Trousers have a single stripe). White tie is a step up (trousers have two stripes just to start).
Then you have morning dress, which gets you a frock coat, waistcoat, top hat, and spats.
It wasn’t that long ago that even a professional (gasp! someone who actually worked for a living) would be expected to wear morning dress when dealing with clients. Doctors most certainly.

The wealthy would have servants whose job it was to ensure the right dress was available at the right time and to get the wearer in and out at the appropriate times as the day progressed. The idle rich could be kept engaged with all manner of frippery. Kept them away from the drinks cabinet.

For the OP, one doubts that the precise technical details matter. Use of a slang term, no matter what it described, would have been the point. Poking fun at such pretensions is a well worn path. The British gave us the TV sitcom Keeping Up Appearances.
Hyacinth Bucket (pronounced bouquet) epitomised the pretentious British middle class. The idea has a long history.

if it was merely the slang ‘tux’ she was upset about, wouldn’t she have told him to use ‘tuxedo’? I feel like she’s upset about something more than just the slang term.

Not always.

Most tuxedo jackets have satin-facing lapels, meaning the lapels are lined with a soft and velvet-like fabric. In comparison, dinner jackets may have traditional lapels without satin, or they may have similar satin-facing lapels as their tuxedo jacket counterparts.

AFAICT from my extremely cursory research, the satin accoutrements gained populartiy in the 1920s-30s. The book was published in 1922, so it’s possible satin hadn’t made its way to Zenith yet.

I came to post that, too! The depth and breadth of general (and often obscure) knowledge possessed by Dopers always astounds and impresses me. I am dazzled.

Me:
:star_struck:

The Master speaks:

Cecil, for example, was recently invited to a bash with a nautical theme advertised as “creative black tie.” Half the guys showed up in tuxedo jackets and jams, but Cecil stole the show with white Izod shorts, Topsiders, tuxedo shirt, $6 black studs and cufflinks, white tuxedo jacket with black pseudo-velvet trim ($7 from the Salvation Army), all topped off with matching sky-blue bowtie and cummerbund featuring palm trees and sailboats. If I had been any cooler I would have been assumed bodily into heaven.

Anyway, I still say Mrs. Babbitt’s problem was U vs. non-U terminology. Vance Packard says so in The Status Seekers:

I’m inclined to agree–she doesn’t tell him to wear a tux instead of a dinner jacket, she tells him not to say ‘tux’, so it seemed to me the issue was specifically with his word choice.